Category Theory: The Beginner’s Introduction

Playlist: https://www.youtube.com/playlist?list=PLm_IBvOSjN4zthQSQ_Xt6gyZJZZAPoQ6v

Video 1	Notes	https://youtu.be/P6DvIfTJhx8
Video 2	Notes	https://youtu.be/4fggq5U3Khg
Video 3	Notes	https://youtu.be/hYO14y50Uso
Video 4	Notes	https://youtu.be/80bbALpA8k8
Video 5	Notes	https://youtu.be/Z-qL2G60zJk
Video 6	Notes	https://youtu.be/TsogM9CEbUk

$Category Music (S) • Objects ○ Pitch Classes § X={x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_10,x_11 } ○ Pitch-Class Names § Z={C♮,C♯,D♭,D♮,D♯,E♭,E♮,F♮,F♯,G♭,G♮,G♯,A♭,A♮,A♯,B♭,B♮} ○ Letter Names of the Pitch-Class Names § L={C,D,E,F,G,A,B} • Arrows ○ n:Z→X § Assign to each name its pitch class ○ t:Z→L § Give the letter name of each pitch-class name ○ i:L→Z § Represents the Major Mode ○ j:L→Z § Represents the Minor Mode Defining Composition • Recall the composition part in the definition of Category ○ Given two arbitrary arrows A →┴f B_1, and B_2 →┴g C ○ We can form the composite A→┴(g∘f) C (called g following f) iff B_1=B_2 ○ e.g. in the case where we have A→┴f B→┴g C • We can rename the composite r and redraw the diagram • But we have to state r=g∘f, because there could be many arrows A→C • To indicate that r=g∘f, we can simply say that this diagram commutes • A Commutative Diagram in any Category is one in which all paths between two objects must be interpreted as the same arrow Composition: Abstract Example • Let s say we have the following three Sets A,B and C ○ A={a_0,a_1 } ○ B={b_0,b_1,b_2 } ○ C={c_0,c_1 } • And the following two maps ○ f:A→B≡{█(f(a_0 )=b_0@f(a_1 )=b_1 )┤ ○ g:B→C≡{█(g(b_0 )=g(b_1 )=c_0@g(b_2 )=c_1 )┤ • We can form g∘f:A→C (which we renamed r), by first applying f then applying g ○ r(a_0 )=(g∘f)(a_0 )=g(f(a_0 ))=g(b_0 )=c_0 ○ r(a_1 )=(g∘f)(a_1 )=g(f(a_1 ))=g(b_1 )=c_0 Composition in Music (S) • We have maps i:L→Z, j:L→Z, we also have a map n:Z→X. So we have • We have the C Minor Scale as a map j:L→Z and the map t:Z→L, so t∘j:L→L exists • So t∘j:L→L equals 1_L:L→L, because t and j have a special relationship$

Category Theory – Video 4

$The Identity Arrow • In S, the Identity Arrow takes each element to itself • e.g. for A={a_0,a_1,a_2 },1_A:A→A≡{█(1_A (a_0 )=a_0@1_A (a_1 )=a_1@1_A (a_2 )=a_2 )┤ • In Music (S), we officially have three more arrows ○ 1_X:X→X ○ 1_Z:Z→Z ○ 1_L:L→L The Identity Laws • Suppose we have a category with objects A and B, then we have 1_A:A→A, 1_B:B→B • The Identity Laws can now be restated to say that, in any Category, this Diagram must Commute • For a diagram to commute, all paths between two objects must be interpreted as the same arrow • Example ○ Suppose f(a_0 )=b_0, then ○ (f∘1_A )(a_0 )=f(1_A (a_0 ))=f(a_0 )=b_0 ○ (1_B∘f)(a_0 )=1_B (f(a_0 ))=1_B (b_0 )=b_0 The Category of Sets With an Endomorphism S^↺ • A^(↺1_A ) and B^(↺1_B ) are objects in S^↺, but what shoud the arrows be? • We want every f:A→B in S to be an arrow f:A^(↺1_A )→B^(↺1_B ) in S^↺ • The law f must follow in S: f∘1_A=1_B∘f, can we generalize this? • What if we replaced 1_A with any endomorphism α, and 1_B with β • Then we get a map f:A^(↺1_A )→B^(↺1_B ) in S^↺ that satisfy: f∘α=β∘f The Associative Law • We have four objects, A,B,C, and D, with arrows f:A→B, g:B→C, and h:C→D • The associativity law says that this diagram must commute • We have 3 paths A→D. They are h∘(g∘f),(hg)∘f, and h∘g∘f • They all must be interpreted as the same map A→D • Example ○ A={a_0,a_1 }, B={b_0,b_1,b_2 }, C={c_0,c_1 },D={d_0,d_1,d_2 } ○ f:A→B≡{█(f(a_0 )=b_0@f(a_1 )=b_1 )┤ ○ g:B→C≡{█(g(b_0 )=g(b_2 )=c_1@g(b_1 )=c_0 )┤ ○ h:C→D≡h(c_0 )=h(c_1 )=d_1$

Shawn Zhong

Shawn Zhong

Category Theory

范畴论 Category Theory

Category Theory: The Beginner’s Introduction

Category Theory – Video 1

Category Theory – Video 2

Category Theory – Video 3

Category Theory – Video 4

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